When is a scheme a quotient of a smooth scheme by a finite group?
Anton Geraschenko, CalTech
Abstract: If a scheme $X$ is a quotient of a smooth scheme by a finite group, it has quotient singularities---that is, it
is \emph{locally} be a quotient by a finite group. In this talk, we will see that the converse is true if $X$ is
quasi-projective and is known to be a quotient by a torus (e.g. $X$ a simplicial toric variety). Though the proof is
stack-theoretic, the construction of a smooth scheme $U$ and finite group $G$ so that $X=U/G$ can be made explicit purely
scheme-theoretically.
To illustrate the construction, I'll produce a smooth variety $U$ with an action of $G=\mathbb Z/2$ so that $U/G$ is the
blow-up of $\mathbb P(1,1,2)$ at a smooth point. This example is interesting because even though $U/G$ is toric, $U$
cannot be taken to be toric.